POSITION / LENGTH /DISTANCE / DISPLACEMENT [metre m]

POSITION / LENGTH /DISTANCE / DISPLACEMENT [metre m] scalar or vector L d dD h H r R a x x y s sxsy nano 1 nm = 10-9 m / micro 1 m = 10-6 m / 1 mm = 10-3 m / 1 km = 103 m sx m1 N +y m2 sy s > 0 s < 0 r -x   +x Gravitation (0,0) +x W E 0 frame of reference m distance travelled d S h displacement magnitude s / direction  N of E acceleration a = constant work = change on KE average velocity average speed u m v uniform circular motion planet M R r Instantaneous velocity R slope = rise / run s rise Kepler’s 3rd law run centripetal force centripetal acceleration t pendulum Length contraction L L0 v L small amplitude only measure L & T g

scalar TIME TIME INTERVAL [second s] t t T ti or t1 initial time time interval s > 0 s < 0 -x +x final time tf or t2 0 frame of reference t =tf – ti = t2 – t1 acceleration a = constant periodT time for one complete oscillation or orbit frequencyf [hertz Hz] number of oscillations (orbits) in one second impulse = change on momentum angular frequency  [rad.s-1] rate at which angle is swept out m u v t  Time Dilation Kepler’s 3rd law pendulum v t0 L planet M R r small amplitude only measure L & T g

slow fast VELOCITY [m.s-1] velocity (vector) / speed (scalar) [1D] v u v0 v1 v2vA VB v > 0 v < 0 [2D] v vxvy vy + v  vx s a/t graph area = v a v > 0 v = 0 Motion with uniform acceleration [1D] v = u + ats = ut + ½ at2v2 = u2 + 2asvav = s/t = (u + v)/2 [2D] horizontal motion ax = 0 vx = uxsx = vxt vertical motion ay = g vy = uy + ay t sy = uyt + ½ay t2 vy2 = uy2 + 2 ay sy v v < 0 t t s/t graph slope tangent = v Object of mass m moving with velocity momentum kinetic energy m Newton’s 1st law Newton’s 2nd law

ACCELERATION [m.s-2] acceleration due to gravity g = 9.8m.s-2 a aav ax ay g ac vector acceleration is the time rate of change of the velocity * getting faster * getting slower * change in direction a(t1) = slope of tangent at t1 v Motion Map t1 t v = area under a / t graph a v Motion with uniform acceleration [1D] v = u + ats = ut + ½ at2v2 = u2 + 2asvav = s/t = (u + v)/2 [2D] horizontal motion ax = 0 vx = uxsx = vxt vertical motion ay = g vy = uy + ay t sy = uyt + ½ay t2 vy2 = uy2 + 2 ay sy t Horizontal & Vertical motions are independent

ACCELERATION [m.s-2] Newton’s 1st Law inertia Newton’s 2nd Law m direction of acceleration same as net force acting on object Centripetal acceleration ac always directed towards centre of circle Some of the effects of acceleration we are familiar with include: Experience of sinking into the seat as a plane accelerates down the runway. "flutter" in our stomach when a lift suddenly speeds up or slows down. “being thrown side ways” in a car going around a corner too quickly. Newton’s 1st Law - inertia

Motion Graphs Instantaneous velocity s = area under v / t graph v slope = rise / run s s t rise run t uniform acceleration a = constant Ball thrown vertically

PROJECTILE MOTION Galileo's analysis of projectile motion: Projectile motion was consisted of both horizontal and vertical components - these components were independent of each other occurred simultaneously perpendicular to each other. Vertical acceleration was the same for all falling objects if air resistance is disregarded. Trajectory of a projectile is a parabola. A motion of an object is relative to its frame of reference and an object has the motion of its inertial frame of reference. He tested this in his Crow's Nest experiment - It was thought that, if a ship was moving at a constant speed, and a ball was dropped from the crows nest, it would fall behind the ship and into the sea as the ship would have moved. Instead, it fell straight down onto the ship as if it hadn't moved. Problem A volcanothat is 3300 m above sea level erupts and sends rock fragments hurting into the sea 9.4 km away. If the fragments were ejected at an angle of 35o, what was their initial speed?

Solution Identify / setup v0= ?m.s-1  = 35o x0= 0 y0= 0 x= 9400 m y= -3300 m ax= 0 ay= -9.8 m.s-2 v0x= v0cos v0y= v0sin Execute X motion Y Motion Eliminate t to find equation for v0 Equation for uniformly accelerated motion how to approach the problem Evaluate The rocks in a volcanic explosion can be thrown out at enormous speeds.

FORCE [newton N] push / pull / interaction between objects F FGW FNN Fff T R vector vector Newton’s 2nd Law Newton’s 1st Law inertia direction of acceleration same as net force acting on object Person is hit by a speeding truck, the magnitude of the forces experienced by the person & truck are the same. Newton’s 3rd Law Newton’s Law of Gravitation WeightFG= W= mg TensionFT = T g = |g| = 9.8 m.s-2 m weight of an object is due to the force acting on it in a gravitational field (attraction between object and planet)

FORCE [newton N] push / pull / interaction between objects F FGW FNN Fff T R vector Friction force Ff = f acts along (parallel) surface Normal force FN = N acts at right angles to a surface Centripetal force Fc

FORCE [newton N] push / pull / interaction between objects F FGW FNN Fff T R vector donkey is the system Free Body Diagram Take the donkey to be the system: – consider only the forces acting on the donkey Coulomb’s Law: force between two pint charges qA & qB Force F on a charged particle q in an electric field E

MOMENTUM p PIMPULSE J[N.s kg.m.s-1] vector For a systemin which the net force acting on it is zero, then the total momentum of the system does not change: Newton’s 3rdLaw  Law of Conservation of Momentum. This law is useful for making predictions when collisions or explosions occur.

KINETIC ENERGY KKEEK / WORK W[ joule J ] scalar Work done W by a single force acting on an object of mass m m moving object has kinetic energy Net force acting over a displacement on object does work on object and changes its kinetic energy:Net Work = Change in kinetic energy More than a single force acting on the object then: Freely falling object: Work done W by the gravitational force FG as an object falls a height h near the surface of the Earth W = FscosF = FG = mg s = h= 0 W = mgh Gravitational potential energy UG = Ep = - W Loss in GPEEp = - mgh m h

Numerical example In golf, a typical contact time is 1.0 ms. If a 45 g ball leaves the club at a speed of 240 km.h-1, estimate the average force exerted by the club on the ball. Solution Identify / Setup t = 1.0 ms = 10-3 s m = 45 g = 4510-3 kg v1 = 0 m.s-1 v2 = 240 km.h-1 = (240)(103)/(3.6103) m.s-1 v2 = 67 m.s-1 Favg = ? N Execute Evaluate units ok significant figures (2) ok Answers approximately equivalent to the mass of 60 people of mass 50 kg Impulse = change in momentum how to approach the problem

Numerical example A boy pushes a toy car of mass 250 g, initially at rest with a horizontal force of 5.0 N through a distance of 1.2 m. How much work is done on the car? What is the final speed of the car? Solution Identify / Setup Execute m = 0.250 g = 0.25 kg Fx= 5.0 N x= 1.2 m W = ? J v2= ? m.s-1v1 = 0 m.s-1 W = Fx x = (5.0)(1.2) J = 6.0 J Evaluate units ok significant figures (2) ok Answers seems OK Work done produces a change in KE W = Fxx = ½ m v22 – ½ mv12 how to approach the problem

GRAVITATION WeightFG= W= mg g = |g| = 9.8 m.s-2 m weight of an object is due to the force acting on it in a gravitational field (attraction between object and planet) Newton’s Law of Gravitation at the Earth’s surface g = 9.8 m.s-2

GRAVITATIONAL POTENTIAL ENERGY EPU UG[joule J] Lifting an object vertically produces an increase in gravitational potential energy of the system of the object and the Earth. Work done = Increase in gravitational potential energy near the Earth’s surface: g = constant g = 9.8 m.s-2(positive number) r   EP  0 Work done W on object in moving it from r to r =  object mass m moved at constant velocity from position r to  to increase its gravitational potential energy final position at r =  m planet M R r initial position at r r   EP  0 gravitational potential energy Ep increases as the distance r from the centre of the planet increases

MOTION OF ROCKETS Rocket launch & propulsion Forces experienced by astronauts during take-off Newton’s third law: Forces act for time interval  impulse: Impulse = Change in momentum: Momentum is conserved: Gases expelled  mass of the rocket decreases  acceleration increases

MOTION OF ROCKETS Force experienced by an astronaut during a space flight Pilots experience large g-forces when entering and pulling out of a steep dive

MOTION OF ROCKETS Earth’s orbital motion around the Sun can be helpful in launching rockets to planets in our Solar System Rocket launch & orbital motion of the Earth Rockets are launched in an easterly direction to get a velocity booster as a result of the Earth spinning about its axis of rotation SLINGSHOT EFFECT - performed to achieve an increase in speed and/or a change of direction of a spacecraft as it travels around a planet. As it approaches, it is caught by the gravitational field of the planet, and swings around it, the speed acquired throws the spacecraft back out again, away from the planet. By controlling the approach, the outcome of the manoeuvre can be manipulated and the spacecraft gain some of the planet’s momentum, relative to the Sun. Simple view from solar reference frame: energy & momentum are conserved http://www.physics.usyd.edu.au/teach_res/hsp/u5/t5_slingshot.htm

MOTION OF ROCKETS - ESCAPE VELOCITY vesc Cannon ball launched from top of a mountain with increasing velocity The escape velocity vescfor a rocket fired from a planet or moon (mass M, radius R) vesc(Earth) = 11 km.s-1 vesc(Moon) = 2.4 km.s-1 vesc(Sun) = 620 km.s-1 R total mechanical energy is conserved E = KE + PE = constant E1 = E2 M orbital velocity

Safe re-entry and landing of rockets or spacecraft • The safe return of a spacecraft into the Earth’s atmosphere and subsequent descent to Earth requires consideration of two main issues: • (1) How to handle the intense heat generated as the spacecraft enters the Earth’s atmosphere. • (2) How to keep the g-forces of deceleration within safe limits. Spacecraft re-entry and the heating effect in passing through the atmosphere. For a safe landing of a spacecraft it must enter the atmosphere in the correct range of angles.

Safe re-entry and landing of rockets or spacecraft • What Can Go Wrong? • If the angle of re-entry is too shallow, the spacecraft may skip off the atmosphere. The commonly cited analogy is a rock skipping across a pond. If the angle of entry is too steep, the spacecraft will burn up due to the heat of re-entry. • Because of collisions with air particles and the huge deceleration, a huge amount of thermal energyis produced from friction. The space shuttle must be able to withstand these temperatures. It uses a covering of insulating tiles which are made of glass fibres but are about 90% air. This gives them excellent thermalinsulating properties and also conserves mass. The tile construction is denser near the surface to make the tiles more resistant to impact damage, but the surface is also porous. Damage to the space shuttle Columbia's heat shield is thought to have causedits disintegration and the loss of seven astronauts on 1st February 2003. Investigators believe that the scorching air of re-entry penetrated a cracked panel on the left wing and melted the metal support structures inside. • Large g-forces are experienced by astronauts as the space shuttle decelerates and re-enters the Earth's atmosphere. Astronauts are positioned in a transverse position with their backs towards the Earth's surface as g-forces are easier for humans to tolerate in these positions. Supporting the body in as many places as possible also helps to increase tolerance. • There is an ionisation blackout for the space shuttle of about 16 minutes where no communication is possible. This is because as thermal energybuilds up, air becomes ionised forming a layer around the spacecraft. Radio signals cannot penetrate this layer of ionised particles.

MOTION OF SATELLITES Satellites are placed in one of several different types of orbit depending on the nature of their mission • Low Earth Orbit (LEO) • Radius: 200 km to 2000 km above the Earth’s surface. • Period: 60 min 90 minutes. Frequent coverage of specific or varied locations on the Earth’s surface. • Small field of view. • Orbits less than 400 km are difficult to maintain due to atmospheric drag and subsequent orbital decay. • Types of satellites: military applications, Earth observation, weather monitoring , shuttle missions. • With the exception of the lunar flights of the Apollo program, all human spaceflights have taken place in LEO. • All manned space stations and the majority of artificial satellites in LEO. • Geostationary Orbit(GEO) • Circular orbit in the Earth's equatorial plane, any point on which revolves about the Earth in the same direction and with the same period as the Earth's rotation. • Useful because they cause a satellite to appear stationary with respect to a fixed point on the rotating Earth. As a result an antenna can point in a fixed direction and maintain a link with the satellite. • The satellite orbits in the direction of the Earth's rotation, at an altitude of approximately 35,786 km above ground. This altitude is significant because it produces an orbital period equal to the Earth's period of rotation, known as the sidereal day. • These orbits allow for the tracking of stationary point on Earth. • Have the largest field of view. • Applications include communications, mass-media and weather monitoring. Orbital decay - reduction in the height of an object's orbit over time due to the drag of the atmosphere on the object. All satellites in low Earth orbits are subject to some degree of atmospheric drag that will eventually decay their orbit and limit their lifetimes. Even at 1000 km, as ‘thin’ as the atmosphere is, it is still sufficiently dense to slow the satellite down over a period of time.

MOTION OF SATELLITES uniform circular motion A force (centripetal force Fc) acting towards the centre of a circle is necessary for an object to rotate in a circular path. Satellite is acted upon by the gravitation force between the Earth and the satellite. The centripetal force is the gravitational force. For a mass m, moving at a speed v, in uniform circular motion of radius r, the net force acting on is called the centripetal force Fcand its magnitude is given by Centripetal force  change in direction of the object as its speed is constant. The resulting acceleration due to the change in direction is the centripetal acceleration acand its magnitude is Directionof the centripetal force and centripetal acceleration is towards the centre of the circle (centripetal means ‘centre seeking’).

MOTION OF SATELLITES ORBITAL MOTION Orbital velocity vorb To place a satellite of mass m into a stable Earth orbit at a particular radius r, the launch must give it both an initial vertical and horizontal component of velocity relative to the Earth’s surface. The satellite will eventually turn so that it is travelling horizontal to the Earth’s surface. At this radius r, the force of gravity FG provides the acceleration needed to keep the object moving in a circle, but a particular orbital velocity is also required to keep the object in a stable orbit – orbital velocity vorb. To calculate that orbital velocity, we equate the centripetal force Fc and gravitational force FG. period T = 24 h Geostationary Orbit m r Orbital velocity of a satellite as it orbits around the Earth only depends on: • mass of the Earth ME • radius of the orbit r Altitude is the only variable that determines the orbital velocity required for a specific orbit around the Earth. Greater the radius of that orbit, the lower that velocity vorb r   vorb  Mplanet net force is always directed to centre of orbit Orbital velocity around other planets MEMplanet

MOTION OF SATELLITES How do the planets move ? Kepler’s Laws of Motion The motion of a planet is governed by the Law of Universal Gravitation F= GMSm/ r2 where Gis the Universal Gravitational Constant, MSis the mass of the Sun, mis the mass of the planet and ris the distance from the Sun to the planet. G= 6.6710-11N.m2.kg2 MS= 2.01030kg Kepler’s 1st law: path of a planet around the Sun is an ellipse. Kepler's Laws of Planetary Motion 1 The path of each planet around the Sun is an ellipsewith the Sun at one focus. 2 Each planet moves so that all imaginary lines drawn from the Sun to the planet sweeps out equal areas in equal periods of time. 3 The ratio of the squares of the periods of revolution of planets is equal to the ratio of the cubes of their orbital radii (mean distance from the Sun or length of semi-major axis, a) Kepler’s 2nd law: Planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time. Computer simulation of the motion of a planet around the Sun. http://www.physics.usyd.edu.au/teach_res/hsp/u5/ag_kepler1.htm

MOTION OF SATELLITES Kepler’s Laws of Periods period T = 24 h = (24)(3.6x103) s = 8.6x104 s Geostationary orbit (GEO) m RE = 6.38x106 m ME = 5.97x1024 kg G = 6.673x10-11 N.m2.kg-1 vorb = ? m.s-1 r = ? m height about Earth’s surface h = ? m h = r - RE m net force is always directed to centre of orbit h r r RE ME M time for one complete orbit: period T distance travelled in one orbit: circumference 2  r Putting in the numbers Rearranging gives Kepler’s Law of Periods r = 4.22x107 m h = 3.59x107 m = 36x103 km vorb = 3.07x103 m.s-1 =1.11x104 km.h-1

EINSTEIN THEORY OF SPECIAL RELATIVITY: SPACE & TIME FRAMES OF REFERENCE Which is bigger – the bear or mouse? – depends upon the location of the observer What is the trajectory of the ball? – it is relative – it depends upon the motion of the observer • Non-Inertial Frames of Reference • * Does not have a constant velocity. It is accelerating. • * The frame could be travelling in a straight line, but be speeding up or slowing down. • * The frame could be travelling along a curved path at a steady speed. • * The frame could be travelling along a curved path and also speeding up or slowing down. • Inertial frame of reference • * Fame of reference with constant velocity. • * Is a non-accelerating frame of reference. • * Law of inertia holds. • * Newton's laws of motion hold. • * No fictitious forces arise.

EINSTEIN THEORY OF SPECIAL RELATIVITY: SPACE & TIME AETHER MODEL FOR THE TRANSMISSION OF LIGHT Classical picture for the speed of light. The speed of light is relative to the motion of the observer, and so the speed of light is c+v or c-v. But this is not correct. The correct answer, is that the person will measure the speed of light to be the constant value c and it does not matter how fast or slower they are approaching or receding from the light beam or the speed of the light source. It seemed inconceivable to 19th Century physicists that light and other electromagnetic waves, in contrast to all other kinds of waves, could propagate without a medium. It seemed to be a logical step to postulate such a medium, called the aether (or ether), even though it was necessary to assume unusual properties for it, such as zero density and perfect transparency, to account for its undetectability. This aether was assumed to fill all space and to be the medium with respect to which electromagnetic waves propagate with the speed c. It followed, using Newtonian relativity, that an observer moving through the aether with velocity v would measure a velocity for a light beam of (c + v). If the aether exists, an observer on Earth should be able to measure changes in the velocity of light due to the Earth’s motion through the aether. The Michelson-Morley experiment attempted to do just this. AETHER – proposed medium for the propagation of electromagnetic waves

EINSTEIN THEORY OF SPECIAL RELATIVITY: SPACE & TIME MICHELSON – MORLEY EXPERIMENT The Michelson-Morley experiment is an excellent example of a critical experiment in science - the fact that no motion of the Earth relative to the aether was detected suggested quite strongly that the aether hypothesis was incorrect and that no aether (absolute) reference frame existed for electromagnetic phenomena – this opened the way for a whole new way of thinking that was to be proposed by Albert Einstein in his Theory of Special Relativity. The null result of the Michelson-Morley experiment was such a blow to the aether hypothesis in particular and to theoretical physics in general that the experiment was repeated by many scientists over more than 50 years. A null result has always been obtained. A null result has always been obtained The light from reflected by the two mirrors produces an interference pattern at the location of the observer. initial interference pattern As the Earth revolves around the Sun and spins on its axis, the direction of the light beams varies with the direction of flow through the aether, their relative velocities would alter and thus the difference in time required for each beam to reach O would alter. This would result in a change in the interference pattern as the apparatus was rotated (changes in the patterns of bright and dark fringes). interference pattern after rotation of apparatus

EINSTEIN THEORY OF SPECIAL RELATIVITY: SPACE & TIME In 1905, Albert Einstein (1879 – 1955) published his famous paper entitled: “On the Electrodynamics of Moving Bodies”, in which he proposed his two postulates of relativity and from these derived his Special Relativity Theory. 1. The Principle of Relativity – All the laws of physics are the same in all inertial reference frames – no preferred inertial frame exists. 2. The Principle of the Constancy of the Speed of Light– the speed of light in free space has the same value c, in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light. The speed of light is constant no matter what are the speeds of the transmitter or receiver. constant c both space and time must be relative quantities Einsteinconcluded - if we accept that the principle of relativity can never be violated, then 1 The aether model must be wrong. 2 The speed of light is constant regardless of the motion of the observer. In order to satisfy, speed of light is constant, he made a revolutionary statement: it is not the speed of light that is changing, but time. Stationary observers and the moving observers perceive space and time differently. In classical physics space and time are constants and motion is defined by them. In Einstein's physics it is the speed of light that is constant and space and time change to accommodate this. Using these ideas, Einstein put forward his Special Theory of Relativity 1 All motion is relative — the principle of relativity holds in all situations. 2 The speed of light is constant regardless of the observer's frame of reference. 3 The aether is not needed to explain light, and, in fact it does not exist.

EINSTEIN THEORY OF SPECIAL RELATIVITY: SPACE & TIME Relativity of simultaneity- whether two spatially separated events occur at the same time is not absolute, but depends on the observer's reference frame. - it is impossible to say in an absolute sense whether two distinct events occur at the same time if those events are separated in space. Observer in train observes train at rest v = 0 Ground based observer sees train go past at speed v v Light beams travel from centre of train and when they hit the ends of the carriage a light flash is given out. Ground based observer sees a flash at the back of the carriage before the flash at the front of the carriage. Train observer sees flashes when light reaches the ends of the carriage simultaneously. http://faraday.physics.utoronto.ca/GeneralInterest/Harrison/SpecRel/Flash/Simultaneity.html

EINSTEIN: SPECIAL RELATIVITY – SPACE AND TIME SPEED OF LIGHT IS CONSTANT – INDEPENDENT OF THE MOTION OF SOURCE OR OBSERVER c = 3.0x108 m.s-1 TIME DILATION proper time t0– the time interval between two events occurring at the same point in space w.r.t. a clock at rest w.r.t. that point. dilated time intervals tv– they are the time intervals on moving clocks w.r.t. a stationary observer. All time intervals measured on moving clocks are longer compared with the stationary clock moving clocks run slower Time Dilation v t0 tv time interval measured by observing moving clock t0 time interval measured by observing stationary clock Time is a relative quantity: different observers can measure different time intervals between the occurrence of two events. This arises because the speed of light is a constant and independent of the motion of the source of light or the motion of an observer. An observer watching a moving clock sees the passage of time on the moving clock to be slower than the passage of time on their own clock. Time intervals are not absolute. This is a violation of a fundamental concepts in Newtonian physics where time is an absolute quantity.

EINSTEIN: SPECIAL RELATIVITY – SPACE AND TIME SPEED OF LIGHT IS CONSTANT – INDEPENDENT OF THE MOTION OF SOURCE OR OBSERVER c = 3.0x108 m.s-1 TIME DILATION Time Dilation v t0 Consider a train with velocity v= 0.90c w.r.t. a stationary frame of reference. In the stationary frame of reference, the duration of an event was 1.00 s. What would be the duration of the event as measured by an observer watching the moving clock? To an observer on Earth, the time taken for the event is 2.29 s. The observer in the train, measures a time interval of only 1.00 s. The Earth observer sees that the train clock has slowed down. It is essential that you understand that this is not an illusion. It makes no sense to ask which of these times is the “real” time. Since no preferred reference frame exists all times are as real as each other. They are the real times seen for the event by the respective observers. Time dilation tells us that a moving clock runs slower than a clock at rest by a factor of 1/{1 – (v2/c2)}. This result, can be generalised beyond clocks to include all physical, biological and chemical processes. The Theory of Relativity predicts that all such processes occurring in a moving frame will slow down relative to a stationary clock. v = 0.90c t0 = 1.00 s tv = ? s moving frame of reference stationary frame of reference

EINSTEIN: SPECIAL RELATIVITY – SPACE AND TIME SPEED OF LIGHT IS CONSTANT – INDEPENDENT OF THE MOTION OF SOURCE OR OBSERVER c = 3.0x108 m.s-1 LENGTH CONTRACTION Length contraction L measured by observer in stationary frame of reference by observing moving object. Lv is the proper length as measured by an observer who is at rest to the object. L0 v L How long is a train? It depends on the relative motion of the observer and the train. This is a real difference in length of the object when it is motion relative to an observer. For a person in the train, there is no contraction in length. v L0 Lv = ? moving frame of reference stationary frame of reference Lv = ?

EINSTEIN: SPECIAL RELATIVITY – SPACE AND TIME LENGTH CONTRACTION / TIME DILATION / MUON DECAY Muons are unstable particles with a rest mass of 207 times that of an electron and a charge of ±1.6x10-19 C. Muonsdecay exponentially into electrons or positrons with a half-life of t1/2 = 1.56x10-6 sas measured in their frame of reference. When high energy particles such as protons called cosmic rays enter the atmosphere from outer space, they interact with air molecules in the upper atmosphere at a height of about 10 km, creating a cosmic ray shower of particles including muons that reach the Earth’s surface. The muons created in these cosmic ray showers travel at v = 0.98c w.r.t to the Earth. Exponential decay: At t = 0 N0 particles After time t, N particles remaining decay constant  = loge(2) / t1/2 Special relativity – length contraction Muon’s frame of reference, the distance from upper atmosphere to Earth’s surface contracted L0 = 10x103 m v = 0.98c Lv = ? m Newtonian (classical) point of view Half-life t1/2 = 1.56x10-6 s Decay constant  = loge(2) / t1/2 = 4.44x105 s-1 Speed of muonsv = 0.98c = (0.98)(3.0x108) m.s-1 = 2.94x108 m.s-1 Distance travelled by muons to reach Earth’s surface = 10x103 m Time to reach Earth’s surface t = 10x103 / 2.94x108 s = 3.40x10-5 s Percentage of muons reaching Earth’s surface = (100)(N/N0) = 100 e-t = (100){exp[(-(4.44x105)(3.4x10-5)]} = 0.00 % Hence, from a Newtonian point of view, most muonswould not be able to reach the Earth’s surface from the upper atmosphere where they are produced. However, experiments show that a large number of muons do reach the Earth’s surface in cosmic ray showers. Time for muons to reach Earth’s surface t = (1.99x103 / 2.94x108) s = 6.77x10-6 s Percentage of muons reaching Earth’s surface = (100)(N/N0) = 100 e-t = (100){exp[(-(4.44x105)(6.77x10-6)]} = 5 % Many many more muons can reach the Earth’s surface then predicted by Newtonian physics  have to reject Newtonian physics and accept Einstein’s postulates: space and time are not absolute quantities.

EINSTEIN: SPECIAL RELATIVITY – SPACE AND TIME MUON DECAY moun clock time interval as measured by Earth observer t0 = 6.77x10-6 s Muons are unstable particles with a rest mass of 207 times that of an electron and a charge of ±1.6x10-19 C. Muonsdecay exponentially into electrons or positrons with a half-life of t1/2 = 1.56x10-6 sas measured in their frame of reference. When high energy particles such as protons called cosmic rays enter the atmosphere from outer space, they interact with air molecules in the upper atmosphere at a height of about 10 km, creating a cosmic ray shower of particles including muons that reach the Earth’s surface. The muons created in these cosmic ray showers travel at v = 0.98c w.r.t to the Earth. v = 0.98c Exponential decay: At t = 0 N0 particles After time t, N particles remaining decay constant  = loge(2) / t1/2 Earth based clock time interval as measured by Earth observer tv = 3.40x10-5 s Special relativity – time dilation Earth observer: muons travels a distance 10 km at a speed 0.98c. Time to reach Earth’s surface tv= 10x103 / 2.94x108 s = 3.40x10-5 s Moving clock’s run slow – observed time interval on muon’s clock is t0 Newtonian (classical) point of view Half-life t1/2 = 1.56x10-6 s Decay constant  = loge(2) / t1/2 = 4.44x105 s-1 Speed of muonsv = 0.98c = (0.98)(3.0x108) m.s-1 = 2.94x108 m.s-1 Distance travelled by muons to reach Earth’s surface = 10x103 m Time to reach Earth’s surface t = 10x103 / 2.94x108 s = 3.40x10-5 s Percentage of muons reaching Earth’s surface = (100)(N/N0) = 100 e-t = (100){exp[(-(4.44x105)(3.4x10-5)]} = 0.00 % Hence, from a Newtonian point of view, most muonswould not be able to reach the Earth’s surface from the upper atmosphere where they are produced. However, experiments show that a large number of muons do reach the Earth’s surface in cosmic ray showers. Percentage of muons reaching Earth’s surface = (100)(N/N0) = 100 e-t = (100){exp[(-(4.44x105)(6.77x10-6)]} = 5 % Same answer as using length contract Note: had to find t0 and not tv.

MASS ENERGY E = mc2 RELATIVISTIC MASS mv / m0 Newtonian mechanics, if a force is applied, an object will accelerate and its velocity will increase indefinitely. Special Relativity, as the velocity approaches the speed of light, the same force produces less and less acceleration, gradually reducing to zero. This is because the mass of the object is increasing and acceleration and mass are inversely proportional when the force is constant (F = ma). If this did not happen the velocity would become infinite. Where does this extra mass come from? The applied force is still doing work (W = Fs) so the object is gaining kinetic energy (since v is increasing). This additional energy is converted into mass according to the equation E = mc2, rather than continually increasing the velocity of the object. Newtonian physics ok v / c mv= measured mass of the moving object mo = measured mass of the object at rest (rest mass) v = relative velocity between the observed object and the observer c = speed of light The mass of an object is a relative quantity, it depends on the relative velocity of the object w.r.t. an observer.

MASS ENERGY E = mc2 Einstein’s famous equation E = mc2  equivalence of mass and energy Energy can be converted into mass and vice versa: * A particle and its antiparticle collide, all the mass is converted into energy. * Mass is converted into energy in a nuclear fissionreaction. * When a body gives off energy Ein the form of radiation, its mass decreases by an amount equal to E/c2. In Special Relativity, the Law of Conservation of Energy and the Law of Conservation of Mass have been replaced by the Law of Conservation of Mass-Energy. When mass increases as a body gains velocity effectively limits all man-made objects to travel at speeds approaching the speed of light. The closer a body gets to the speed of light, the more massive it becomes. The more massive it becomes,the more energy that has to be used to give it the same acceleration. To accelerate the body up to the speed of light would require an infinite amount of energy. Clearly, this places a limit on both the speed that can be attained by a spacecraft and therefore the time it takes to travel from one point in space to another.

POSITION / LENGTH /DISTANCE / DISPLACEMENT [metre m]